User:Jrm228/Benchmark IV: Discussion

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Author: James McGinnity

Discussion: Modeling Excitatory and Inhibitory Neurons

Conclusions

Model vs. Hypothesis

In our bifurcation analysis, outlined in the Results section, we were able to demonstrate the behavior of both Izhikevich's original and the Hodgkins-Huxley model demonstrated in Chapter 7 of our textbook. There were considerable differences between the nullclines of the membrane potential, and the value of the equilibrium points of the models were not the same, either. The hypothesis that we were testing was that the Izhikevich did not lose a significant amount of information when he simplified the Hodgkins-Huxley model to be computationally efficient, and, based on the nullclines and equilibrium points, our model does not support our hypothesis. It should be noted that this is in spite of the replications of both models being incredibly visually similar.

Limitations

There are several inherent limitations of our results that we found difficult or impossible to overcome. First, the two models that we were comparing had a different set of initial parameter values and were constructed differently. The Izhikevich model was meant to be simpler, and so it only had two state variables compared to the four state variables in the Chapter 7 model of the Hodgkins-Huxley equations. These state variables provide the depth of detail that leads to our dismissal of our hypothesis, and yet it is hard to say whether or not the systems would be comparable if we had the same number of state variables.

In addition, as outlined in our model description, there are several assumptions we had to make about both systems that undermine the applicability of our data to real life systems.

Model Discrepancies

Our results section outlined in greater detail the discrepancies between our replications and those originally presented by Izhikevich. In his paper, the code he used artificially spiked the neurons to 30 mV every time to show consistency, while we simply increased the number of data points to have our graph show 30 mV spikes every time. In addition, we used a single set of equations with changing parameters based on what he provided in the paper. However, Izhikevich changed his equations every time he made a graph in order to increase the accuracy of his work.

Literature Review of Conclusions

Once we had reached the conclusion that our hypothesis was not supported by our models, we needed to determine why the models did not match each other. In one of our cited sources from the introduction, the processes that control the neuron's membrane potential include a series of equations governing the exchange of ions across the membrane during firing [1][2]. The concentration of potassium ions, sodium ions, and other materials changes constantly through the spiking, creating the differences in charge that force the spike. However, in the model created by Izhikevich, these are represented as a series of parameters that are unique to a firing period [3]. Their static state creates graphs that look very similar for each individual scenario, but do not provide the same level of detail as the Hodgkins-Huxley model from Chapter 7, which had a state variable for each of the main ion contributors in the system [4]. These concentration gradients are incredibly important to the membrane potential, and are the most likely reason, according to literature, that the two models have such starkly different nullclines in our bifurcation analysis.

Future Work

A reasonable extension of this work beyond the bifurcation analysis that we conducted would be to see exactly what information was lost between the two models. From our nullcline results, it would appear that the primary loss of detail came in the current that was injected into the system, which could be potentially corrected to provide the biologically plausable and computationally efficient model that Izhikevich wanted to provide. If that proved to be impossible to represent, we could also try to go with our original plan with regards to the comparison, which was to create a network of thousands of spiking cortical neurons in place of a bifurcation analysis to show an increasing level of difference between the systems as the number of neurons increased.

References

  1. Hagan, Martin; Howard, Demuth; Beale, Mark; De Jess, Orlando (2014). Neural Network Design (2nd ed.). New York: Martin Hagan. 
  2. Rojas, Raul (2013). Neural Networks: A Systematic Introduction (1st ed.). Berlin: Springer. 
  3. Izhikevich, Eugene. "Simple Model of Spiking Neurons". IEEE Transactions on Neural Networks. 14 (6): 1569–1572. 
  4. Chiel, Hillel J. (2017). Chapter 7 (18th ed.). Hillel J. Chiel.